Hillary Clinton on Saturday will pledge to make an overhaul of the campaign-finance system an urgent priority, as she announces plans to introduce a constitutional amendment overturning the Supreme Court’s Citizens United ruling during the first 30 days of her administration.

The presumptive Democratic nominee has made her opposition to the Supreme Court decision lifting restrictions on corporate and union political expenditures a prominent feature in her campaign. And on Saturday, Mrs. Clinton will sketch out how she will...